Dynamical systems · smooth perturbation theory · periodic orbits

Unrestricted C^r Closing Lemma for r ≥ 2

Collaboration beta

A recurrent orbit returns arbitrarily close to itself. The question is whether an arbitrarily small unrestricted C^r change can always close such recurrence into a periodic orbit when r is at least 2. The classical C^1 theorem does not provide the general higher-regularity result.

g_{n} \to f, u_{n} \to x, g_{n}^{m_{n}} (u_{n}) = u_{n}

Known results and sources

The unrestricted C^r closing problem asks whether every recurrence can be made periodic by an arbitrarily small perturbation when r is at least 2.

Research problem

Exact mathematical statement

Let $M$ be a compact smooth manifold, $f\in\operatorname{Diff}^r(M)$ with integer $r\ge2$ , and $x$ a nonwandering or recurrent point of $f$ . Prove that every $C^r$ neighborhood of $f$ contains a diffeomorphism $g$ for which $x$ is periodic, or equivalently at vanishing cost, a sequence of nearby diffeomorphisms with periodic points converging to $x$ that can be moved to $x$ by vanishing local conjugacies. Symbolically, the desired nearby periodic form is

g_n\to f,\qquad u_n\to x,\qquad g_n^{m_n}(u_n)=u_n.

The perturbations are unrestricted in $C^r$ . A theorem confined to a conservative, symplectic, Hamiltonian, fixed-flux, fixed-energy, KAM, or other constrained category does not settle this statement.

Problem infographic

Problem at a glance

Problem-first closing-lemma diagram showing recurrence, the desired arbitrarily small perturbation, the distinction between C^1 and restricted high-regularity results, and the unresolved higher-regularity bridge. — The classical C^1 theorem and restricted C^r results are separated from the open unrestricted r-at-least-2 problem and its source-reported derivative and width barriers.

Current mathematical picture

Where work on Unrestricted C^r Closing Lemma for r ≥ 2 stands

Open problem

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureSuperseded finite-scale gateways

The source’s supersession map records explicit failure modes and directs future work to the dual-matched seed, global cone patching, exact terminal branches, and recentered resource accounting. The exact two-point bridge and robust-width or top-jet targets remain viable if their perturbation budgets and point alignment are proved.

Route status · Narrowed route

Main reductionDeep branch pair terminalizes

Exact inverse branches convert a deep opposed pair into either a shorter periodic point or a shorter return with small singular defect.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeConvert the two-point derivative bridge into usable pointwise or orbit-flat data.Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Unrestricted C^r Closing Lemma for r ≥ 2 in numbers

2.7kretained lines of mathematical investigation2,700 in the current working snapshot

Argument development: 2,210 · 82%
Explored or eliminated routes: 136 · 5%
Computational analysis: 37 · 1%
Open obligations: 102 · 4%
Definitions and setup: 215 · 8%

8selected mapped statements1routes investigated4open questions4contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Status checkedAug 14, 2026

Source recordedAug 14, 2026

Recommended next task

Convert the two-point derivative bridge into usable pointwise or orbit-flat data.

Suggested move: Identify the normalized tensor carried by rho times D-squared S and prove a second localization, a cheap quadratic or projective correction, or a strict improvement in residual-to-radius or robust capacity.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded

Narrowed routeSuperseded finite-scale gateways

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks

Convert the two-point derivative bridge into usable pointwise or orbit-flat data.Suggested move: Identify the normalized tensor carried by rho times D-squared S and prove a second localization, a cheap quadratic or projective correction, or a strict improvement in residual-to-radius or robust capacity.

Ready to work on

Establish robust-width descent in the r equals 2 near-saturation regime.Suggested move: Use controlled adjacent radius ratios and least-return geometry to force comparable close branches, a telescoping nested chain, or enough comparable levels for endpoint allocation.

Ready to work on

Robust-width descent or root

After the derivative bridge, width collapse must yield descent, branch intersection, a top-jet root, or a finite normalized obstruction.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Prove a one-dimensional reduced top-jet root theorem with exact regularity.Suggested move: Write the scalar reduced return germ on a terminal interval and force a zero by degree or sign while controlling lower terms, endpoint motion, and the terminal periodic representative.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026

Current statusOpen problem

The C^1 theorem is classical, and high-regularity closing statements are known in important constrained or typical settings. The unrestricted C^r diffeomorphism statement for every integer r at least 2 remains open.

[2][5][4]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2026
Peer reviewedXue developed a KAM-normal-form approach and proved typical-perturbation conclusions in specific settings, described as partial progress on the high-regularity problem.[5]
2023
PreprintShi and Wang proved a C^r chain-closing result for a specified partially hyperbolic class.[4]
2006
Peer reviewedXia and Zhang proved a C^r closing lemma for a restricted class of partially hyperbolic symplectic diffeomorphisms.[3]
1983
Peer reviewedPugh and Robinson gave the C^1 closing lemma in a framework including Hamiltonian systems.[2]

Mathematical neighborhood

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

6 standing statements2 proposed statements4 open questions1 narrowed routes

Statements by mathematical role8 selected mapped statements

theorem candidate1 of 81
reduction4 of 84
lemma2 of 82
negative result1 of 81

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.21 displayed rows · 1 route included

retained route statementCan arbitrarily small unrestricted C^r perturbations close every recurrence?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementExact recurrent-point formintermediate
retained route statementRestricted results do not settle itintermediate
retained route statementEndpoint failure gives low gainintermediate
retained route statementDeep branch pair terminalizesintermediate
retained route statementTwo-point derivative dichotomyintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureSuperseded finite-scale gatewaysreported failure
Research targetConvert the two-point derivative bridge into usable pointwise or orbit-flat data.open
Research targetEstablish robust-width descent in the r equals 2 near-saturation regime.open
Research targetProve a one-dimensional reduced top-jet root theorem with exact regularity.open
Research targetRobust-width descent or rootopen
Narrowed routeSuperseded finite-scale gatewaysThe source’s supersession map records explicit failure modes and directs future work to the dual-matched seed, global cone patching, exact terminal branches, and recentered resource accounting. The exact two-point bridge and robust-width or top-jet targets remain viable if their perturbation budgets and point alignment are proved.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeConvert the two-point derivative bridge into usable pointwise or orbit-flat data.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

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Survive an independent attempt to falsify the proposed step.

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Prepared starting pointConvert the two-point derivative bridge into usable pointwise or orbit-flat data.

Unrestricted C^r Closing Lemma for r ≥ 2 · ready to start

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Sources and references5 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

1
The Closing Lemmapeer reviewed result · Charles C. Pugh · American Journal of Mathematics · 1967 · DOI 10.2307/2373413 · accessed Aug 14, 2026
2
The C^1 Closing Lemma, including Hamiltonianspeer reviewed result · Charles C. Pugh, Clark Robinson · Ergodic Theory and Dynamical Systems · 1983 · DOI 10.1017/S0143385700001978 · accessed Aug 14, 2026
3
A C^r Closing Lemma for a Class of Symplectic Diffeomorphismspeer reviewed result · Zhihong Xia, Hua Zhang · Nonlinearity · 2006 · ARXIV math/0503225 · DOI 10.1088/0951-7715/19/2/015 · accessed Aug 14, 2026
4
C^r-chain closing lemma for certain partially hyperbolic diffeomorphismspreprint · Yi Shi, Xiaodong Wang · arXiv · 2023 revision · ARXIV 2210.15896 · accessed Aug 14, 2026
5
Closing Lemma and KAM Normal Formpeer reviewed result · Jinxin Xue · Acta Mathematica Sinica, English Series · 2026-04-17 · ARXIV 2207.06208 · DOI 10.1007/s10114-026-4598-7 · accessed Aug 14, 2026

Important qualifications

The review is statement-scoped to compact smooth manifolds and unrestricted C^r perturbations of diffeomorphisms for every integer r at least 2.
Hamiltonian, symplectic, KAM, typical-perturbation, chain-closing, and partially hyperbolic results are retained only as restricted neighbors.
Scoped searches of current mathlib and Isabelle public documentation found no end-to-end formalization of the unrestricted statement; that does not establish absence.

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Unrestricted C^r Closing Lemma for r ≥ 2

Exact mathematical statement

Problem at a glance

Where work on Unrestricted C^r Closing Lemma for r ≥ 2 stands

Unrestricted C^r Closing Lemma for r ≥ 2 in numbers

Convert the two-point derivative bridge into usable pointwise or orbit-flat data.

How the current approaches connect

Research route map

Other routes

Open questions

The known mathematical landscape

What the literature has established

Related results and reusable starting points

Formalization opportunities

What changed in the research record

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Contribute

Important qualifications

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